Algebraic & Geometric Topology

Euclidean Mahler measure and twisted links

Daniel S Silver, Alexander Stoimenow, and Susan G Williams

Full-text: Open access


If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, not necessarily alternating, have bounded twist numbers, then both the Jones polynomials and a parametrization of the 2–variable Homflypt polynomials of the corresponding links have bounded Mahler measure.

Article information

Algebr. Geom. Topol., Volume 6, Number 2 (2006), 581-602.

Received: 26 March 2005
Accepted: 15 March 2006
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 37B40: Topological entropy

link twist number Alexander polynomial Jones polynomial Mahler measure


Silver, Daniel S; Stoimenow, Alexander; Williams, Susan G. Euclidean Mahler measure and twisted links. Algebr. Geom. Topol. 6 (2006), no. 2, 581--602. doi:10.2140/agt.2006.6.581.

Export citation


  • S Bhatty, Mahler measure and the Alexander polynomial of pretzel links, Undergraduate honors thesis (S G Williams, advisor) University of South Alabama, Mobile AL (2004)
  • D W Boyd, Speculations concerning the range of Mahler's measure, Canad. Math. Bull. 24 (1981) 453–469
  • J W Brown, R V Churchill, Complex variables and applications, McGraw-Hill Book Co., New York (1984)
  • P J Callahan, J C Dean, J R Weeks, The simplest hyperbolic knots, J. Knot Theory Ramifications 8 (1999) 279–297
  • A Champanerkar, I Kofman, On the Mahler measure of Jones polynomials under twisting, Algebr. Geom. Topol. 5 (2005) 1–22
  • A Champanerkar, I Kofman, E Patterson, The next simplest hyperbolic knots, J. Knot Theory Ramifications 13 (2004) 965–987
  • P R Cromwell, Homogeneous links, J. London Math. Soc. $(2)$ 39 (1989) 535–552
  • O Dasbach, X-S Lin, A volume-ish theorem for the Jones polynomial of alternating knots
  • G Everest, T Ward, Heights of polynomials and entropy in algebraic dynamics, Universitext, Springer London Ltd., London (1999)
  • C M Gordon, Toroidal Dehn surgeries on knots in lens spaces, Math. Proc. Cambridge Philos. Soc. 125 (1999) 433–440
  • E Hironaka, The Lehmer polynomial and pretzel links, Canad. Math. Bull. 44 (2001) 440–451
  • V F R Jones, }, Ann. of Math. $(2)$ 126 (1987) 335–388
  • E Kalfagianni, Alexander polynomial, finite type invariants and volume of hyperbolic knots, Algebr. Geom. Topol. 4 (2004) 1111–1123
  • K Kodama, Knot, a program for computing knot invariants, available at\char'176\penalty -100 kodama/knot.html
  • M Lackenby, The volume of hyperbolic alternating link complements, Proc. London Math. Soc. $(3)$ 88 (2004) 204–224 (with an appendix by Ian Agol and Dylan Thurston)
  • D H Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. $(2)$ 34 (1933) 461–479
  • D A Lind, T Ward, Automorphisms of solenoids and $p$-adic entropy, Ergodic Theory Dynam. Systems 8 (1988) 411–419
  • K Mahler, On some inequalities for polynomials in several variables, J. London Math. Soc. 37 (1962) 341–344
  • K Murasugi, On alternating knots, Osaka Math. J. 12 (1960) 277–303
  • K Murasugi, 0142116
  • K Murasugi, J H Przytycki, The skein polynomial of a planar star product of two links, Math. Proc. Cambridge Philos. Soc. 106 (1989) 273–276
  • K Murasugi, A Stoimenow, The Alexander polynomial of planar even valence graphs, Adv. in Appl. Math. 31 (2003) 440–462
  • H Seifert, Über das Geschlecht von Knoten, Math. Ann. 110 (1935) 571–592
  • D S Silver, W Whitten, Hyperbolic covering knots, Algebr. Geom. Topol. 5 (2005) 1451–1469
  • D S Silver, S G Williams, Lehmer's question, knots and surface dynamics
  • D S Silver, S G Williams, Mahler measure, links and homology growth, Topology 41 (2002) 979–991
  • D S Silver, S G Williams, Mahler measure of Alexander polynomials, J. London Math. Soc. $(2)$ 69 (2004) 767–782
  • A Stoimenow, Alexander polynomials and hyperbolic volume of arborescent links, preprint
  • A Stoimenow, A property of the skein polynomial with an application to contact geometry
  • A Stoimenow, The Jones polynomial, genus and weak genus of a knot, Ann. Fac. Sci. Toulouse Math. $(6)$ 8 (1999) 677–693
  • A Stoimenow, On the coefficients of the link polynomials, Manuscripta Math. 110 (2003) 203–236